Description
Evaluation code over \(GF(q)\) on a set of points \({\cal P} = \left( P_1,P_2,\cdots,P_n \right)\) in \(GF(q)\) lying on an algebraic curve \(\cal X\) whose corresponding vector space \(L\) of functions \(f\) consists of certain polynomials or rational functions.
Codewords are evaluations of all functions at the specified points, \begin{align} \left( f(P_1), f(P_2), \cdots, f(P_n) \right) \quad\quad\forall f\in L~. \tag*{(1)}\end{align} The code is denoted as \(C_L({\cal X},{\cal P},D)\), where the divisor \(D\) (of degree less than \(n\)) determines which rational functions to use by prescribing features associated with their zeroes and poles. The original motivation for evaluation codes, which are generalizations of RS codes that expand both the types of functions used as well as the available evaluation points, was to increase code length while maintaining good distance and size.
The algebraic curve \(\cal X\) used for this construction is the set of zeroes of one or more nontrivial polynomials forming a prime ideal. Such curves are both smooth and irreducible over any field extension of \(GF(q)\). The curve can be defined over affine space or projective space, which contains the affine coordinates as a subset and which can yield an increase in length. If evaluations are made over projective coordinates, then the codewords are evaluations of homogeneous polynomials, and there are relations between such polynomials with polynomials over affine space. See Refs. [1–4] for more details.
Protection
A lower bound on distance is \(d \geq n- \deg (D)\) [5; Thm. 15.3.12]. The order or Feng-Rao bound, a generalization of the shift bound for cyclic codes, gives a lower bound on the distance of evaluation AG codes [6–8]. Connection to semigroups yields another bound [3,9].Decoding
Generalization of plane-curve decoder [10]. Another decoder [11] was later showed to be equivalent in Ref. [12]. Application of several algorithms in parallel can be used to decode up to half the minimum distance [13,14]. Computational procedure implementing these decoders is based on an extension of the Berlekamp-Massey algorithm by Sakata [15–17].Decoder based on majority voting of unknown syndromes [6] decodes up to half of the minimum distance [18].List decoders generalizing Sudan's RS decoder by Shokrollahi-Wasserman [19] and Guruswami-Sudan [20].Cousins
- Linear \(q\)-ary code— The degree of the divisor for evaluation AG codes is restricted to be less than \(n\). When there is no restriction, any \(q\)-ary linear code can be formulated as an evaluation AG code [24].
- Frameproof (FP) code— Asymptotic bounds on FP codes can be formulated using evaluation AG codes [25,26]. A sufficient condition for an evaluation AG code to be FP can be recast as an instance of the Riemann-Roch equation [27; Sec. 15.8.2].
- Quantum AG code
Primary Hierarchy
Parents
Evaluation AG codes are evaluation codes for which \(\cal X\) is an algebraic curve, i.e., an algebraic variety of dimension one [3].
Evaluation AG code
Children
Elliptic codes are evaluation AG codes with \(\cal X\) being an elliptic curve, i.e., curve of genus one [1,3][2; Ch. 3.2].
Norm-trace codes are evaluation AG codes with \(\cal X\) being a Miura-Kamiya curve [28].
Suzuki-curve codes are evaluation AG codes with \(\cal X\) being a Suzuki curve.
GRS (RS) codes are in one-to-one correspondence with evaluation AG codes of univariate polynomials \(f\) with \(\cal X\) being the projective (affine) line [1,3][5; Thm. 15.3.24][2; Ch. 3.2].
Any residue AG code of differential forms can be equivalently stated as an evaluation AG code of functions [1,2][5; Lemma 15.3.10][3; Thm. 2.72]. In addition, evaluation and residue AG codes are dual to each other [3][5; pg. 313].
TVZ codes are evaluation AG codes where \(\cal X\) is either a Drinfeld modular curve, a classic modular curve, or a Garcia-Stichtenoth curve, but can also be formulated as residue AG codes.
Tamo-Barg-Vladut codes are evaluation AG codes on algebraic curves, such as Hermitian or Garcia-Stichtenoth curves.
The hexacode is an evaluation AG code over the quaternary Galois field \(GF(4) = \{0,1,\omega, \bar{\omega}\}\) with \(\cal X\) defined by \(x^2 y + \omega y^2 z + \bar{\omega} z^2 x = 0\) [3; Exam. 2.77].
References
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Page edit log
- En-Jui Kuo (2024-03-18) — most recent
- Victor V. Albert (2024-03-18)
- Victor V. Albert (2022-08-11)
- Victor V. Albert (2022-03-22)
Cite as:
“Evaluation AG code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/evaluation